Ricci dynamo stretch-shear plasma flows and magnetic energy bounds

TL;DR

This paper applies geometrical tools from Einstein's general relativity to dynamo theory, deriving bounds on magnetic energy using Ricci flows and symmetries, with implications for plasma flows and magnetic field amplification.

Contribution

It introduces a novel approach combining Ricci flow geometry and Killing symmetries to establish bounds on magnetic energy in plasma dynamos, extending previous hydromagnetic results.

Findings

Magnetic field aligns with shear flow tensor eigendirection in marginal dynamos.

Killing symmetries simplify the computation of energy bounds.

Ricci curvature acts analogously to diffusion in dynamo dynamics.

Abstract

Geometrical tools, used in Einstein's general relativity (GR), are applied to dynamo theory, in order to obtain fast dynamo action bounds to magnetic energy, from Killing symmetries in Ricci flows. Magnetic field is shown to be the shear flow tensor eigendirection, in the case of marginal dynamos. Killing symmetries of the Riemann metric, bounded by Einstein space, allows us to reduce the computations. Techniques used are similar to those strain decomposition of the flow in Sobolev space, recently used by Nu\~nez [JMP \textbf{43} (2002)] to place bounds in the magnetic energy in the case of hydromagnetic dynamos with plasma resistivity. Contrary to Nu\~nez case, we assume that the dynamos are kinematic, and the velocity flow gradient is decomposed into expansion, shear and twist. The effective twist vanishes by considering that the frame vorticity coincides with Ricci rotation coefficients. Eigenvalues are here Lyapunov exponents. In analogy to GR, where curvature plays the role of gravity, here Ricci curvature seems to play the role of diffusion.